ISEM 2016: Ecological Network Analysis Workshop
Bipartite Networks
M.K. Lau (Harvard Forest, Harvard University)
Updated “12 May 2016”
Bipartite Networks
- Community ecology’s gateway to ecological networks
Bipartite Networks: Community ecology’s gateway to ecological networks
- Bipartite: a network of interacting organisms where interactions are represented only between two distinct groups and not within these two groups.
Bipartite Networks from Community Matrices
Community Matrix = Occurrences (Sites x Species)
| sp1 | sp2 | sp3 | |
|---|---|---|---|
| site1 | 1 | 1 | 1 |
| site2 | 1 | 1 | 0 |
| site3 | 1 | 1 | 0 |
| site4 | 1 | 0 | 0 |
| site5 | 1 | 0 | 0 |
Bipartite Networks from Community Matrices
Bipartite Network = Interactions (Species x Species)
| sp6 | sp7 | sp8 | |
|---|---|---|---|
| sp1 | 1 | 1 | 1 |
| sp2 | 1 | 1 | 0 |
| sp3 | 1 | 1 | 0 |
| sp4 | 1 | 0 | 0 |
| sp5 | 1 | 0 | 0 |
Bipartite Networks
Bipartite Networks
Examples:
- Plant Pollinator
- Plant Herbivore
- Plant Parasite
Mutualistic Networks (Bascompte and Jordano 2014)
Individuals
- In reality, interactions occur among individuals
Individuals <-> Genotype <-> Species
Genotype or other genetic clustering information can help to bridge the gap between looking at interactions among all individuals and integrating to the species level.
Genotype-Species (Lau et al. 2016)
In general, Bipartite Networks
- Advantage: Provides a theoretical framework that provides mechanistic hypotheses for stability
- Limitation: a sub-set of the entire community and interactions
Bipartite Network Analysis
The following notes focus on using the bipartite package for analysis and draws directly from this paper:
Dormann, C.F., Fruend, J., Bluethgen, N. & Gruber B. 2009. Indices, graphs and null models: analyzing bipartite ecological networks. The Open Ecology Journal, 2, 7-24.
Bipartite Network Metrics
- There are many ways to look at a bipartite network
- These metrics can be categorized into metrics derived directly from the network graph information (base quantities) and metrics that combine the lower level metrics together (higher order indices)
Base Quantities
- L: number of realized links in a network
- I: number of lower trophic level species (e.g. plants, prey)
- J: number of higher trophic level species (e.g. pollinators, parasitoids)
- m: total number of interactions for all species
- \(a_{ij}\): number of interactions between species i from the lower and species j from the higher trophic level, also referred as “link weight”
- \(A_i\) (\(\sum_{j=1}^J a_{ij}\)): total number of interactions of species i from the lower trophic level
- \(A_j\) (\(\sum_{i=1}^I a_{ij}\)): total number of interactions of species j from the higher trophic level
Higher Indices (Unweighted)
- Links per Species
- Connectance
- Cluster Coefficient
- Web Asymmetry
- Number of Compartments
- Extinction Slopes
- Degree Distribution
- Mean Number of Shared Hosts
- Togetherness
- C-Score
- V-Ratio
- Nestedness
Higher Indices (Weighted)
- Nestedness (WINE)
- Modularity
- Generality
- Vulnerability
- Weighted Linkage Density
- Interaction Evenness
- Interaction Strength Asymmetry
- Niche Overlap
The main focus of bipartite ecological networks has been:
- Nestedness
- Modularity
Nestedness
Nestedness: Packing Communities
Nestedness is the tendancy for interactions in communities to be organized such that species with fewer interactions interact with subsets of species with more interactions.
This is suggested to facilitate community stability by decreasing competition for resources and optimizing the redundancy of interactions.
Nestedness: Packing Communities
Measuring Nestedness: Many Metrics
1. Ordering
2. Isocline
3. Deviations
Exercise: Play with matrix ordering
Exercise: Play with matrix ordering
M <- c(rep(1,10),
c(rep(1,4),rep(0,6)),
c(rep(1,3),rep(0,7)),
c(rep(1,2),rep(0,8)),
rep(c(rep(1,1),rep(0,9)),6)
)
M <- matrix(M,nrow=10)
rownames(M) <- paste('P',1:10,sep='')
colnames(M) <- paste('A',1:10,sep='')Exercise: Play with matrix ordering
M## A1 A2 A3 A4 A5 A6 A7 A8 A9 A10
## P1 1 1 1 1 1 1 1 1 1 1
## P2 1 1 1 1 0 0 0 0 0 0
## P3 1 1 1 0 0 0 0 0 0 0
## P4 1 1 0 0 0 0 0 0 0 0
## P5 1 0 0 0 0 0 0 0 0 0
## P6 1 0 0 0 0 0 0 0 0 0
## P7 1 0 0 0 0 0 0 0 0 0
## P8 1 0 0 0 0 0 0 0 0 0
## P9 1 0 0 0 0 0 0 0 0 0
## P10 1 0 0 0 0 0 0 0 0 0Exercise: Play with matrix ordering
M[sample(1:10,10),sample(1:10,10)]## A7 A9 A4 A3 A2 A10 A1 A5 A6 A8
## P8 0 0 0 0 0 0 1 0 0 0
## P2 0 0 1 1 1 0 1 0 0 0
## P7 0 0 0 0 0 0 1 0 0 0
## P5 0 0 0 0 0 0 1 0 0 0
## P4 0 0 0 0 1 0 1 0 0 0
## P1 1 1 1 1 1 1 1 1 1 1
## P6 0 0 0 0 0 0 1 0 0 0
## P10 0 0 0 0 0 0 1 0 0 0
## P3 0 0 0 1 1 0 1 0 0 0
## P9 0 0 0 0 0 0 1 0 0 0Exercise: Play with matrix ordering
M[sample(1:10,10),sample(1:10,10)]## A3 A9 A10 A6 A5 A8 A4 A2 A1 A7
## P2 1 0 0 0 0 0 1 1 1 0
## P1 1 1 1 1 1 1 1 1 1 1
## P3 1 0 0 0 0 0 0 1 1 0
## P9 0 0 0 0 0 0 0 0 1 0
## P4 0 0 0 0 0 0 0 1 1 0
## P6 0 0 0 0 0 0 0 0 1 0
## P5 0 0 0 0 0 0 0 0 1 0
## P7 0 0 0 0 0 0 0 0 1 0
## P8 0 0 0 0 0 0 0 0 1 0
## P10 0 0 0 0 0 0 0 0 1 0Exercise: Play with matrix ordering
M[sample(1:10,10),sample(1:10,10)]## A10 A1 A2 A9 A8 A3 A5 A4 A6 A7
## P8 0 1 0 0 0 0 0 0 0 0
## P9 0 1 0 0 0 0 0 0 0 0
## P5 0 1 0 0 0 0 0 0 0 0
## P3 0 1 1 0 0 1 0 0 0 0
## P4 0 1 1 0 0 0 0 0 0 0
## P7 0 1 0 0 0 0 0 0 0 0
## P1 1 1 1 1 1 1 1 1 1 1
## P10 0 1 0 0 0 0 0 0 0 0
## P2 0 1 1 0 0 1 0 1 0 0
## P6 0 1 0 0 0 0 0 0 0 0Testing for Nestedness
- Calculate the nestedness for a given network
- Choose an algorithm to shuffle
- Shuffle the interactions
- Calculate nestedness for the shuffled network
- Repeat a lot, like thousands of times
- Compare the observed to the “null” nestedness values
This is a general recipe for bipartite network analysis
- Calculate the metric for a given network
- Choose an algorithm to shuffle
- Shuffle the interactions
- Calculate metric for the shuffled network
- Repeat a lot, like thousands of times
- Compare the observed to the “null” metric values
Testing for Nestedness
obs <- 45
null <- rnorm(1000,65,9)
hist(null,main='')
abline(v=obs)
Testing for Nestedness
z.score <- (obs - mean(null)) / sd(null)
p.value <- length(null[null <= obs])/length(null)
z.score## [1] -2.180544p.value## [1] 0.011
Exercise: Nestedness Analysis
library(bipartite)
data(Safariland)
SafarilandExercise: Nestedness Analysis
## Policana albopilosa Bombus dahlbomii
## Aristotelia chilensis 673 0
## Alstroemeria aurea 0 154
## Schinus patagonicus 0 0
## Berberis darwinii 0 67
## Rosa eglanteria 0 0
## Cynanchum diemii 0 0
## Ribes magellanicum 0 0
## Mutisia decurrens 0 0
## Calceolaria crenatiflora 0 0
## Ruizantheda mutabilis Trichophthalma amoena
## Aristotelia chilensis 110 0
## Alstroemeria aurea 0 0
## Schinus patagonicus 0 0
## Berberis darwinii 0 0
## Rosa eglanteria 6 0
## Cynanchum diemii 0 0
## Ribes magellanicum 0 2
## Mutisia decurrens 0 0
## Calceolaria crenatiflora 0 0
## Syrphus octomaculatus Manuelia gayi
## Aristotelia chilensis 0 0
## Alstroemeria aurea 5 7
## Schinus patagonicus 0 0
## Berberis darwinii 5 0
## Rosa eglanteria 4 0
## Cynanchum diemii 0 0
## Ribes magellanicum 0 0
## Mutisia decurrens 0 0
## Calceolaria crenatiflora 0 0
## Allograpta.Toxomerus Trichophthalma jaffueli
## Aristotelia chilensis 0 0
## Alstroemeria aurea 1 3
## Schinus patagonicus 0 0
## Berberis darwinii 0 0
## Rosa eglanteria 2 0
## Cynanchum diemii 0 0
## Ribes magellanicum 3 0
## Mutisia decurrens 0 0
## Calceolaria crenatiflora 1 0
## Phthiria Platycheirus1 Sapromyza.Minettia
## Aristotelia chilensis 0 4 0
## Alstroemeria aurea 8 1 1
## Schinus patagonicus 0 0 0
## Berberis darwinii 0 0 0
## Rosa eglanteria 0 0 0
## Cynanchum diemii 0 0 0
## Ribes magellanicum 0 0 0
## Mutisia decurrens 1 0 0
## Calceolaria crenatiflora 0 0 0
## Formicidae3 Nitidulidae Staphilinidae
## Aristotelia chilensis 0 0 0
## Alstroemeria aurea 0 0 4
## Schinus patagonicus 0 0 0
## Berberis darwinii 0 0 0
## Rosa eglanteria 0 0 3
## Cynanchum diemii 8 1 0
## Ribes magellanicum 0 0 0
## Mutisia decurrens 0 0 0
## Calceolaria crenatiflora 0 0 0
## Ichneumonidae4 Braconidae3 Chalepogenus caeruleus
## Aristotelia chilensis 1 0 0
## Alstroemeria aurea 0 0 0
## Schinus patagonicus 15 0 0
## Berberis darwinii 0 0 0
## Rosa eglanteria 0 0 0
## Cynanchum diemii 0 2 0
## Ribes magellanicum 0 0 0
## Mutisia decurrens 0 0 0
## Calceolaria crenatiflora 0 0 3
## Vespula germanica Torymidae2 Phthiria1
## Aristotelia chilensis 0 0 0
## Alstroemeria aurea 4 0 1
## Schinus patagonicus 0 0 0
## Berberis darwinii 0 0 0
## Rosa eglanteria 0 0 0
## Cynanchum diemii 0 9 0
## Ribes magellanicum 0 0 0
## Mutisia decurrens 0 0 0
## Calceolaria crenatiflora 0 0 0
## Svastrides melanura Sphecidae Thomisidae
## Aristotelia chilensis 0 0 0
## Alstroemeria aurea 6 1 1
## Schinus patagonicus 0 0 0
## Berberis darwinii 0 0 0
## Rosa eglanteria 0 0 0
## Cynanchum diemii 0 0 0
## Ribes magellanicum 0 0 0
## Mutisia decurrens 0 0 0
## Calceolaria crenatiflora 0 0 0
## Corynura prothysteres Ichneumonidae2
## Aristotelia chilensis 1 0
## Alstroemeria aurea 3 4
## Schinus patagonicus 0 0
## Berberis darwinii 0 0
## Rosa eglanteria 0 0
## Cynanchum diemii 0 0
## Ribes magellanicum 0 0
## Mutisia decurrens 0 0
## Calceolaria crenatiflora 0 0
## Ruizantheda proxima Braconidae2
## Aristotelia chilensis 0 1
## Alstroemeria aurea 4 0
## Schinus patagonicus 0 0
## Berberis darwinii 0 0
## Rosa eglanteria 0 0
## Cynanchum diemii 0 0
## Ribes magellanicum 0 0
## Mutisia decurrens 0 0
## Calceolaria crenatiflora 0 0Exercise: Nestedness Analysis
plotweb(Safariland)
Exercise: Nestedness Analysis
plotweb(Safariland,text.rot=90)
Exercise: Nestedness Analysis
nested(Safariland)## binmatnest2.temperature
## 19.81353Exercise: Nestedness Analysis
nested(Safariland,method='ALL')## binmatnest2.temperature binmatnest discrepancy2.discrepancy
## 19.7534631 5.7626076 15.0000000
## discrepancy C.score checker
## 21.0000000 0.5616097 1.0227920
## NODF2 NODF weighted NODF
## 24.5478036 18.0878553 11.6063738
## wine
## 0.3685509Exercise: Nestedness Analysis
safari.nest <- nestedness(Safariland)
names(safari.nest)## [1] "mat" "n.rows" "n.cols"
## [4] "temperature" "n.nullmodels" "population.size"
## [7] "n.individuals" "binmatnestout" "n.generations"
## [10] "nullmodels" "p.null1" "mean.temp.null1"
## [13] "var.temp.null1" "p.null2" "mean.temp.null2"
## [16] "var.temp.null2" "p.null3" "mean.temp.null3"
## [19] "var.temp.null3" "pack.order.col" "pack.order.row"
## [22] "packed.matrix"Exercise: Nestedness Analysis
unlist(safari.nest[c('temperature')])## temperature
## 5.762608unlist(safari.nest[c('p.null1','p.null2','p.null3')])## p.null1 p.null2 p.null3
## 0 0 0unlist(safari.nest[c('mean.temp.null1','mean.temp.null2','mean.temp.null3')])## mean.temp.null1 mean.temp.null2 mean.temp.null3
## 28.14982 27.33150 18.72031unlist(safari.nest[c('var.temp.null1','var.temp.null2','var.temp.null3')])## var.temp.null1 var.temp.null2 var.temp.null3
## 27.66088 33.82524 36.37121Modularity
Modularity: Eggs in Multiple Baskets
Nestedness and Modularity: measure interaction structure
Nestedness Revisited
Nestedness and Modularity: measure interaction structure
Nestedness Revisited
Modularity: Eggs in Multiple Baskets
Modularity: Eggs in Multiple Baskets
Modularity: Eggs in Multiple Baskets
- Modularity: the tendancy of species to interact more within than between groups that are subsets of the entire community
- This is thought to stabilize communities by restricting the impact of disturbances within groups and preventing cascading effects throughout the larger community
Modularity: Eggs in Multiple Baskets
Modularity
safari.mods <- computeModules(Safariland)Modularity
plotModuleWeb(safari.mods)
Testing Modularity
null.nets <- lapply(1:5,commsimulator,x=Safariland,method='r1')
null.mods <- lapply(null.nets,computeModules)
null.stats <- lapply(null.mods,function(x) slot(x,'likelihood'))
null.stats <- unlist(null.stats)Testing Modularity
obs.mods <- slot(safari.mods,'likelihood')
null.mu <- mean(null.stats)
null.sd <- sd(null.stats)
null.z <- (obs.mods - null.mu) / null.sd
null.p <- length(null.stats[null.stats <= obs.mods])/length(null.stats)
null.z## [1] -1.291243null.p## [1] 0
Nested and Modules
Further reading on bipartite network analysis
- General: https://goedoc.uni-goettingen.de/goescholar/bitstream/handle/1/5837/Dormann.pdf?sequence=1
- Nestedness: https://www.uvm.edu/~ngotelli/manuscriptpdfs/UlrichConsumersGuide.pdf
- Modularity: http://arxiv.org/pdf/1304.3218.pdf
- Null Models: https://www.uvm.edu/~ngotelli/manuscriptpdfs/UlrichEcology.pdf
- Mutualistic Networks: http://press.princeton.edu/titles/10161.html
- Null Models: http://deploy.extras.ufg.br/projetos/adrimelo/div/Gotelli2000-NullModels.pdf